Calculus-performing mechanical calculator

A clip from the Discovery Channel's Dirty Jobs program on tanneries demonstrates the workings of a calculus-performing mechanical calculator that measures the surface-area of irregularly shaped hides with a fascinating and clever set of gears, calipers and ratchets.

A planimeter, of which I have a couple, is a mechanical device that does more or less the same thing–more slowly and requiring more operator skill. It traces the outside of an irregular shape and computes the area through a hardware–as in wheels, gears, and arms–implementation of Green’s Theorem, an algorithm for computing the double integral of a surface by a path integral of its boundary. Newer planimeters are electronic and not nearly as much fun, or as good to look at. The instruction booklet that came with my K+E planimeter, bought on eBay, actually explains the math of how it works.

This machine looks really ingenious and very accurate, but I don’t think it’s performing calculus, at least not what we commonly consider calculus. It’s measuring discrete strips of area and mechanically adding them together as a sum on the main dial. This machine could probably be used to demonstrate the fundamentals of calculus though. That planimeter described above seems like an implementation of calculus.

I’ve always heard it called the trapezoidal rule, myself. First thing I learned in calculus, only thing I ever needed… just program your computer to start at one and increment the number of trapezoids until the result suddenly goes loony tunes – that’s where you’ve reached the limit of precision of your device, so take the last result before the meltdown and that’s the area under your curve, or close enough for any real-world physical need. You can do this with Simpson’s rule too, I suppose.

No, what the other guy described (the planimeter) is performing calculus. This machine seems to use the ‘Method Of Exhaustion’ to approximate the area of the shape in rectangular strips of discrete width and possibly of discrete increments of height. This makes it a mechanical brute-force computer of area. As someone else pointed out, and as I pointed out in my comment, this machine demonstrates how calculus works, but doesn’t perform it. It doesn’t derive from the curve what the area is.

the planimeter is more elegant and much more precise, but they are both “just measuring” approximations to an integral. operator error and mechanical tolerances are never exact and if you want to get ridiculously pedantic, there is always the planck length…
the method displayed above is much more practical for directly measuring solid objects rather than just closed plane curves.

78 years ago, some disgruntled engineers were grumbling to each other after work about how this device is trivial, and back in their educations/apprenticeships they did basically the same thing with stuff lying around the lab, and if you want to see a real patent, just look at James Watt’s steam engine but man of course he got screwed out of the money, just like every real innovator, and…